Find the asymptotes of the following hyperbolas and also the equations to their conjugate hyperbolas.
$55 x^{2} - 120 x y + 20 y^{2} + 64 x - 48 y = 0 .$

Open in App

Solution

Given equation of the hyperbola: $55 x^{2} - 120 x y + 20 y^{2} + 64 x - 48 y = 0$
The equation of a hyperbola and the combined equation of the asymptotes differ only in the constant term.
Therefore the combined equation of the asymptotes is of the form: $55 x^{2} - 120 x y + 20 y^{2} + 64 x - 48 y + K = 0... (i)$
Factorising the 2nd degree terms, gives
$55 x^{2} - 120 x y + 20 y^{2} = 55 x^{2} - 110 x y - 10 x y + 20 y^{2} = (x - 2 y) (55 x - 10 y)$
Therefore the asymptotes have equations $x - 2 y + l = 0$ and $55 x - 10 y + m = 0$ and their combined equation will be $(x - 2 y + l) (55 x - 10 y + m) = 0... (i i)$ .
Equations (i) & (ii) represent the same pair of lines.
Comparing the coefficient of x, y and constant term gives,
$55 l + m = 64, - 2 m - 10 l = - 48$ & $l m = K$ ,
which gives $l = \frac{4}{5}$ , and $m = 20$
Therefore the asymptotes are $x - 2 y + \frac{4}{5} = 0$ and $55 x - 10 y + 20 = 0$ and their combined equation is $55 x^{2} - 120 x y + 20 y^{2} + 64 x - 48 y + 16 = 0$ .

Now using the fact that: equation of hyperbola + equation of conjugate hyperbola = 2x(equation of asymptotes)
$\Rightarrow$ equation of conjugate hyperbola = 2x(equation of asymptotes) - equation of hyperbola
$\Rightarrow$ equation of conjugate hyperbola = $2 (55 x^{2} - 120 x y + 20 y^{2} + 64 x - 48 y + 16) - (55 x^{2} - 120 x y + 20 y^{2} + 64 x - 48 y)$
$\Rightarrow$ equation of conjugate hyperbola = $55 x^{2} - 120 x y + 20 y^{2} + 64 x - 48 y + 32 = 0$

Suggest Corrections

Similar questions

19 x^{2} + 24 x y + y^{2} - 22 x - 6 y = 0 .

y^{2} - x y - 2 x^{2} - 5 y + x - 6 = 0 .

8 x^{2} + 10 x y - 3 y^{2} - 2 x + 4 y = 2 .

2 x^{2} - 3 x y - 2 y^{2} + 3 x - y + 8 = 0

Find the equation of conjugate hyperbola whose equations of asymptotes are x + 2y + 3 = 0 and 3x + 4y + 5 = 0 and hyperbola passes through (1, -1).

Join BYJU'S Learning Program